Simplify and expand the following expression: $ \dfrac{5z + 9}{z + 1}+\dfrac{z}{z - 3} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(z + 1)(z - 3)$ Multiply the first term by $\dfrac{z - 3}{z - 3}$ $ \begin{align*} \dfrac{5z + 9}{z + 1} \times \dfrac{z - 3}{z - 3} & = \dfrac{(5z + 9)(z - 3)}{(z + 1)(z - 3)} \\ & = \dfrac{5z^2 - 6z - 27}{(z + 1)(z - 3)}\end{align*} $ Multiply the second term by $\dfrac{z + 1}{z + 1}$ $ \begin{align*} \dfrac{z}{z - 3} \times \dfrac{z + 1}{z + 1} & = \dfrac{(z)(z + 1)}{(z - 3)(z + 1)} \\ & = \dfrac{z^2 + z}{(z - 3)(z + 1)}\end{align*} $ Now we have: $ = \dfrac{5z^2 - 6z - 27}{(z + 1)(z - 3)} + \dfrac{z^2 + z}{(z - 3)(z + 1)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{5z^2 - 6z - 27 + z^2 + z}{(z + 1)(z - 3)} $ $ = \dfrac{6z^2 - 5z - 27}{(z + 1)(z - 3)}$ Expand the denominator: $ = \dfrac{6z^2 - 5z - 27}{z^2 - 2z - 3}$